Supplementary material: The increasing cost of happiness

Corresponding author:
Professor Nick Glozier
Faculty of Medicine and Health,
University of Sydney,
NSW 2050,
Australia
email: nick.glozier@sydney.edu.au

Supplementary Material

Sample characteristics

The broad demographic characteristics of the sample at each sixth year wave are presented below in Table S1. The linear trend as a function of time was estimated for each variable (linear regression of time for continuous variables and Cochrane-Armitage test in the case of binary variables), and Bon-Ferroni adjusted p-values for multiple comparisons are presented (k = 11).

Table S1. Demographic characteristics of the sample
Characteristic	2001 N = 12,5341	2007 N = 11,3851	2013 N = 15,4931	2019 N = 15,7901	p-value2
Household income ($)	42,553 (±26,170)	50,528 (±45,267)	55,863 (±33,845)	57,078 (±34,078)	<0.001
Life satisfaction	7.9 (±1.7)	7.9 (±1.5)	7.9 (±1.5)	7.9 (±1.5)	>0.9
Happiness	68.2 (±16.8)	68.4 (±16.7)	68.3 (±16.8)	66.2 (±17.7)	<0.001
Males	5,949 (47%)	5,348 (47%)	7,363 (48%)	7,491 (47%)	>0.9
Age (years)	46.0 (±16.6)	46.9 (±17.4)	47.4 (±17.8)	48.3 (±18.3)	<0.001
University Graduates	2,303 (18%)	2,539 (22%)	4,012 (26%)	4,568 (29%)	<0.001
Unemployed	474 (3.8%)	278 (2.4%)	568 (3.7%)	528 (3.3%)	>0.9
Couples	8,682 (69%)	7,721 (68%)	10,564 (68%)	10,686 (68%)	0.14
Chronic illness	3,128 (25%)	3,315 (29%)	4,974 (32%)	4,865 (31%)	<0.001
SEIFA	5.5 (±2.9)	5.6 (±2.8)	5.6 (±2.8)	5.6 (±2.8)	>0.9
Household size	2.9 (±1.5)	2.8 (±1.4)	2.8 (±1.5)	2.7 (±1.4)	<0.001
1Mean (±SD); n (%)
2linear trend analysis (parametric); Cochrane-Armitage test

Household income and average life satisfaction increased between 2001-2019, while average happiness score decreased slightly over the 19 years. The proportions of each sex were stable over time, as was the proportion of people living as a couple (Couples), average household size and SEIFA index. The percentage unemployed varied with economic conditions however there was no significant positive or negative linear trend. Age and chronic health conditions tended to increase over time. The slight increase in the age of the panel over time (2.3 years) was substantially less than that of a cohort study which would be 18 years. Education levels (i.e., percentage of university graduates) also tended to increase over time.

Generally, Table S1 supports the view that the HILDA sample was a relatively stable representation of Australian socioeconomic conditions between 2001 and 2019. The importance of the specific demographic variables with a significant linear trend (age, age², education, illness) were tested in regression models of happiness as covariates (see below Figure S3).

Model Comparisons

Linear vs non-linear models. We compared the out-of-sample evidence for the non-linear (piecewise) models with the linear models of income on wellbeing in each year, using the difference in WAIC deviance scores (i.e., WAIC_linear — WAIC_piecewise). Thus a WAIC difference greater than zero indicated evidence for a linear relationship; a WAIC difference less than zero indicated evidence for a nonlinear (piecewise) relationship.

Figure S1. Linear model evidence (WAIC_linear — WAIC_piecewise)

Figure S1 legend: Differences in WAIC scores (deviance units) for a linear fit over a piecewise fit for happiness (red) and satisfaction (blue). The filled circle indicates the mean of the difference and the horizontal bars represents the 95% interval. A difference below zero (grey vertical line) is support for the piecewise model (a difference above zero is support for the linear model).

The difference in WAIC scores for each model revealed the nonlinear (piecewise) fits of happiness, but not satisfaction, provided superior out-of-sample accuracy over a linear fit in each year. Figure S1 shows the 95% interval of the differences in WAIC scores clearly distinguished the advantage of the nonlinear fits for happiness with no overlap with zero in any year. By contrast, the difference in WAIC between the linear and nonlinear models of life satisfaction was not distinguished from zero for most years (with the exception of 2017 and 2008), indicating the (out-of-sample) accuracy of the two models was largely indistinguishable over the 19 years. Overall the evidence suggested that household income has a distinct nonlinear relationship with happiness; implying the existence of a change point or difference in slopes at different levels of household income. This is consistent with the only meta-analysis conducted in this area, over ten years ago, which concluded the linear relationship between income and life satisfaction is stronger than that with happiness (Howell and Howell, 2008). It is also worth noting that a linear function between income and life satisfaction is consistent with the majority of research since then (Stevenson and Wolfers, 2013).

Conditional vs unconditional models of happiness. Table S1 (above) showed the specific variables that changed over the 19 years between 2001-2019. In particular, the number of university graduates, average age and the incidence of chronic illness increased slighty over the period. Including these variables as covariates in any model (i.e,. a conditional model such that P( $y$ | income, age, education illness)) will improve the in-sample fit (e.g., R²), and also adjust the parameter estimates of interest (e.g., change point) conditional upon their common association with happiness. Given the general aim of the main paper was to describe the temporal change in functional form between happiness and income in Australia, regardless of any other variable (i.e., an unconditional model of P($y$| income)) or sampling differences, neither the in-sample fit nor the conditional model estimates are directly relevant. Nevertheless, the importance of these variables in mediating the changes we observed in the unconditional model are relevant, and so we calculated the estimates from conditional piecewise models including age, age², sex, education (university degree), chronic illness for each year below to confirm the pattern of temporal change was similar. Figure S2 shows the addition of these covariates did not change the trend in change point parameter estimates (red) relative to the unconditional parameter estimates (grey) over the 19 years.

Figure S2. Conditional model parameters

Figure S2 legend: Comparison of the conditional parameter estimates (red) and unconditional parameter estimates (grey) in the change point model of income on happiness. The conditional models included covariates for age, age squared, sex, education, and illness. Horizontal bar represents the 95% credible region and the solid point indicates the expected value (median) of each distribution.

The parameter estimates of the conditional model (red) were adjusted for the broad demographic variables which tended to increase over the 19 years in our sample (age, education, illness); however the pattern of changes over time was comparable to the unconditional model (grey), indicating these temporal trends were not responsible for the evolution in the change point between income and happiness.

Additional non-linear fits

Log-linear models. The relationship between happiness and income is often described as log-linear (e.g., Kahneman and Deaton, 2010). Under this view, similar proportional changes in income or wealth produce similar evaluations of satisfaction or happiness, rather than the absolute change. That is, a $100 increase in wealth will produce a greater impact on someone earning a minimum wage than someone earning many orders of magnitude of that amount. Likewise, the piecewise model that we employed can also adopt an increasing monotonic function whose slope reduces with wealth, albeit with a single identifiable change-point rather than a smooth curve. However our main thesis is not committed to a function with a single discrete changepoint; only that the slope between income and happiness is unequal at different levels of income, and the function is shifting to the right over time. The choice between functions is somewhat arbitrary, since a log-linear function is as easily consistent with our thesis as the piecewise function we described in the main paper. To resolve this, we compared the evidence for a log-linear fit against the piecewise fits according to their WAIC scores. Thus a WAIC difference less than zero indicated evidence for the piecewise fit over the log-linear fit (i.e., greater out-of-sample accuracy).

Figure S3. Log-linear model evidence (WAIC_log-linear — WAIC_piecewise)

Figure S3 legend: The joint distribution between wellbeing and income for a single year, imposed by the priors.

The piecewise fit was superior to a log-linear fit of income on happiness in each year, although the difference was less distinguishable in 2003, 2006, and 2008 where the 95% interval overlapped zero. This indicates the evidence for a piecewise model was generally greater than for a log-linear model, however it does not necessarily imply the presence of a discrete change point exists. In a Bayesian change point model, the estimated location of the change point in each year is not a single point but is rather a probability distribution of income values that extends over the entire range of household income. Importantly the width of the 95% posterior probability distribution in Figure 2 indicates the change point location is likely over a range of values, on average ±$9.15K, which implies a more or less smooth function over that range.

Prior and residual checks

We used regularizing Guassian priors centred on zero, with Normal(0, 0.1) for each β, and a Normal(0, 0.5) for ω. A plot of the joint distribution between income and wellbeing imposed by the priors alone confirmed there was no apparent slope or change point implied by their regularizing effect.

Figure S4. Prior predictive check

Figure S4 legend: The joint distribution between wellbeing and income for a single year, imposed by the priors. The solid blue line indicates a smoothed average (loess).

Examination of the residual plots from the piecewise models of happiness confirms the residuals were evenly distributed around zero and there was little indication of heteroscedascity. Overall, there was little evidence that the piecewise models of happiness were not appropriate for the data.

Figure S5. Residual plots for piecewise happiness models

Figure S5 legend: The joint distribution between residual and fitted values for each year.

Parameter and fit summary statistics

There were N = 59,876 total observations. 4000 samples (post-warmup) were drawn using sampling(NUTS). For each parameter, Bulk_ESS and Tail_ESS are effective sample size (ESS) measures, and Rhat is the potential scale reduction factor on split chains. In each case the Rhat approached 1, indicating convergance. Each ESS n > 300, which is a conservative threshold for estimation.

Table S2. Piecewise parameter estimates and fit statistics for happiness
Year	Parameter	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
2001	Intercept	0.054	0.030	-0.005	0.113	1.0	1352	1611
	Pre-slope	0.231	0.033	0.173	0.298	1.0	1723	1931
	Post-slope	0.069	0.015	0.038	0.097	1.0	1806	1921
	Change-point	-0.025	0.201	-0.376	0.402	1.0	1287	1662
	sigma	0.992	0.007	0.979	1.005	1.0	2705	2212
2002	Intercept	0.050	0.033	-0.008	0.124	1.0	1537	1399
	Pre-slope	0.248	0.037	0.182	0.329	1.0	1533	1633
	Post-slope	0.072	0.016	0.041	0.102	1.0	1907	1924
	Change-point	-0.068	0.218	-0.424	0.478	1.0	1390	1200
	sigma	0.991	0.007	0.978	1.004	1.0	2282	2340
2003	Intercept	0.087	0.045	0.009	0.181	1.0	1019	1346
	Pre-slope	0.227	0.030	0.175	0.293	1.0	1363	1800
	Post-slope	0.079	0.020	0.037	0.115	1.0	1401	1456
	Change-point	0.202	0.284	-0.245	0.828	1.0	978	1274
	sigma	0.990	0.007	0.976	1.003	1.0	2293	2014
2004	Intercept	0.090	0.036	0.019	0.164	1.0	1429	1317
	Pre-slope	0.254	0.027	0.203	0.309	1.0	2100	1887
	Post-slope	0.086	0.018	0.049	0.120	1.0	1848	1784
	Change-point	0.161	0.190	-0.198	0.570	1.0	1295	1329
	sigma	0.987	0.007	0.973	1.001	1.0	2666	2065
2005	Intercept	0.122	0.038	0.051	0.198	1.0	1294	1584
	Pre-slope	0.258	0.028	0.207	0.315	1.0	1825	2150
	Post-slope	0.062	0.020	0.021	0.098	1.0	1772	2229
	Change-point	0.290	0.208	-0.081	0.718	1.0	1206	1648
	sigma	0.987	0.007	0.974	1.001	1.0	2986	2768
2006	Intercept	0.099	0.032	0.038	0.165	1.0	1426	1761
	Pre-slope	0.293	0.031	0.238	0.357	1.0	1800	2147
	Post-slope	0.074	0.017	0.040	0.105	1.0	1718	2178
	Change-point	0.106	0.165	-0.190	0.459	1.0	1325	1439
	sigma	0.986	0.007	0.973	1.000	1.0	2853	2559
2007	Intercept	0.074	0.024	0.026	0.120	1.0	1436	1631
	Pre-slope	0.367	0.040	0.292	0.451	1.0	2001	2099
	Post-slope	0.066	0.015	0.036	0.094	1.0	1882	2129
	Change-point	-0.141	0.109	-0.355	0.086	1.0	1508	1641
	sigma	0.986	0.007	0.973	1.000	1.0	2850	2465
2008	Intercept	0.068	0.038	0.004	0.161	1.0	1150	767
	Pre-slope	0.313	0.041	0.234	0.396	1.0	1316	1196
	Post-slope	0.083	0.017	0.046	0.114	1.0	1421	1114
	Change-point	-0.049	0.206	-0.345	0.472	1.0	1152	661
	sigma	0.986	0.007	0.973	1.000	1.0	2739	2199
2009	Intercept	0.077	0.034	0.009	0.148	1.0	1424	1669
	Pre-slope	0.274	0.031	0.217	0.341	1.0	2016	2074
	Post-slope	0.081	0.017	0.046	0.113	1.0	1753	1965
	Change-point	0.059	0.189	-0.305	0.462	1.0	1419	1600
	sigma	0.987	0.007	0.974	1.001	1.0	2527	1673
2010	Intercept	0.086	0.033	0.025	0.151	1.0	1311	1884
	Pre-slope	0.306	0.042	0.235	0.392	1.0	1492	1701
	Post-slope	0.063	0.016	0.029	0.093	1.0	1625	1861
	Change-point	0.010	0.188	-0.318	0.390	1.0	1226	1906
	sigma	0.988	0.007	0.975	1.002	1.0	2591	2179
2011	Intercept	0.098	0.026	0.044	0.148	1.0	1184	1538
	Pre-slope	0.323	0.031	0.266	0.388	1.0	1656	1501
	Post-slope	0.060	0.015	0.030	0.088	1.0	1660	2323
	Change-point	0.024	0.128	-0.230	0.294	1.0	1290	1474
	sigma	0.986	0.006	0.974	0.998	1.0	2402	2010
2012	Intercept	0.089	0.032	0.035	0.156	1.0	1185	1666
	Pre-slope	0.302	0.033	0.242	0.369	1.0	1494	2142
	Post-slope	0.078	0.015	0.045	0.106	1.0	1689	1501
	Change-point	0.056	0.171	-0.211	0.422	1.0	1108	1731
	sigma	0.986	0.006	0.974	0.998	1.0	2780	2439
2013	Intercept	0.103	0.028	0.043	0.158	1.0	1369	1551
	Pre-slope	0.245	0.025	0.201	0.296	1.0	1759	1701
	Post-slope	0.061	0.016	0.030	0.092	1.0	1513	1898
	Change-point	0.226	0.161	-0.107	0.559	1.0	1337	1505
	sigma	0.990	0.006	0.978	1.001	1.0	2760	2674
2014	Intercept	0.084	0.029	0.026	0.144	1.0	1302	1630
	Pre-slope	0.281	0.031	0.227	0.345	1.0	1661	2182
	Post-slope	0.071	0.014	0.042	0.099	1.0	1657	2007
	Change-point	0.056	0.162	-0.240	0.402	1.0	1257	1576
	sigma	0.988	0.006	0.977	1.000	1.0	2570	2419
2015	Intercept	0.157	0.028	0.101	0.213	1.0	1253	1433
	Pre-slope	0.252	0.020	0.214	0.294	1.0	1531	1493
	Post-slope	0.036	0.018	0.002	0.071	1.0	1886	2016
	Change-point	0.445	0.154	0.149	0.762	1.0	1073	1036
	sigma	0.988	0.006	0.976	0.999	1.0	3084	2691
2016	Intercept	0.106	0.031	0.051	0.174	1.0	1265	1570
	Pre-slope	0.273	0.029	0.220	0.332	1.0	1599	1927
	Post-slope	0.056	0.016	0.022	0.085	1.0	1748	1939
	Change-point	0.156	0.182	-0.135	0.564	1.0	1145	1604
	sigma	0.989	0.006	0.977	1.001	1.0	2651	2078
2017	Intercept	0.142	0.027	0.087	0.193	1.0	1805	1547
	Pre-slope	0.256	0.022	0.214	0.301	1.0	2110	1778
	Post-slope	0.046	0.015	0.015	0.076	1.0	2056	2616
	Change-point	0.373	0.146	0.082	0.656	1.0	1654	1250
	sigma	0.988	0.006	0.977	1.000	1.0	3009	2422
2018	Intercept	0.138	0.029	0.084	0.198	1.0	1618	1939
	Pre-slope	0.252	0.021	0.212	0.296	1.0	2124	2192
	Post-slope	0.061	0.016	0.028	0.090	1.0	2083	2468
	Change-point	0.381	0.156	0.092	0.716	1.0	1602	2020
	sigma	0.988	0.006	0.976	0.999	1.0	3264	2613
2019	Intercept	0.134	0.041	0.056	0.212	1.0	1171	1520
	Pre-slope	0.221	0.021	0.185	0.268	1.0	1748	1495
	Post-slope	0.067	0.019	0.028	0.102	1.0	1451	2032
	Change-point	0.465	0.245	0.011	0.968	1.0	1147	1382
	sigma	0.989	0.006	0.977	1.000	1.0	2742	2459

Table S3. Linear parameter estimates and fit statistics for life satisfaction
Year	Parameter	Estimate	Est.Error	l-95% CI	u-95% CI	Rhat	Bulk_ESS	Tail_ESS
2001	Intercept	0.001	0.009	-0.016	0.018	1.0	4050	2833
	dollars	0.054	0.009	0.036	0.071	1.0	4279	3161
	sigma	0.999	0.006	0.987	1.011	1.0	4741	2988
2002	Intercept	0.000	0.009	-0.018	0.019	1.0	3543	3103
	dollars	0.047	0.009	0.029	0.065	1.0	4063	3220
	sigma	0.999	0.007	0.986	1.012	1.0	4401	2927
2003	Intercept	0.001	0.010	-0.018	0.020	1.0	4823	3212
	dollars	0.045	0.009	0.027	0.063	1.0	4394	2983
	sigma	0.999	0.007	0.986	1.012	1.0	4616	2769
2004	Intercept	0.001	0.009	-0.018	0.019	1.0	3749	3154
	dollars	0.058	0.009	0.040	0.077	1.0	4510	3263
	sigma	0.999	0.007	0.986	1.012	1.0	4354	2917
2005	Intercept	0.000	0.009	-0.018	0.019	1.0	4075	2814
	dollars	0.050	0.010	0.031	0.070	1.0	3855	2989
	sigma	0.999	0.006	0.987	1.011	1.0	4393	2921
2006	Intercept	0.000	0.009	-0.018	0.019	1.0	4893	3066
	dollars	0.045	0.009	0.027	0.063	1.0	4060	3283
	sigma	0.999	0.007	0.986	1.012	1.0	4859	3270
2007	Intercept	0.000	0.009	-0.018	0.018	1.0	4201	3204
	dollars	0.070	0.009	0.052	0.088	1.0	4111	2728
	sigma	0.998	0.007	0.985	1.011	1.0	4188	2947
2008	Intercept	0.001	0.009	-0.018	0.019	1.0	3472	3128
	dollars	0.065	0.009	0.046	0.083	1.0	3778	3035
	sigma	0.998	0.006	0.986	1.011	1.0	4759	3218
2009	Intercept	0.000	0.009	-0.017	0.018	1.0	3699	3036
	dollars	0.068	0.009	0.050	0.086	1.0	4224	2660
	sigma	0.998	0.006	0.985	1.010	1.0	4992	3119
2010	Intercept	0.001	0.009	-0.017	0.018	1.0	3517	2728
	dollars	0.061	0.009	0.043	0.079	1.0	4155	3378
	sigma	0.998	0.006	0.986	1.011	1.0	4812	3249
2011	Intercept	0.001	0.008	-0.015	0.017	1.0	3961	2898
	dollars	0.086	0.008	0.070	0.103	1.0	3828	2658
	sigma	0.996	0.006	0.986	1.008	1.0	5033	3154
2012	Intercept	0.001	0.008	-0.016	0.016	1.0	3668	2583
	dollars	0.080	0.008	0.064	0.096	1.0	3783	2627
	sigma	0.997	0.006	0.986	1.008	1.0	4026	2925
2013	Intercept	0.001	0.008	-0.015	0.016	1.0	4369	3221
	dollars	0.078	0.008	0.062	0.093	1.0	3968	2994
	sigma	0.997	0.006	0.986	1.008	1.0	5067	2908
2014	Intercept	0.001	0.008	-0.015	0.016	1.0	4072	2858
	dollars	0.099	0.008	0.083	0.114	1.0	3893	2809
	sigma	0.995	0.006	0.984	1.006	1.0	4344	2937
2015	Intercept	0.001	0.008	-0.015	0.017	1.0	3204	2750
	dollars	0.086	0.008	0.071	0.102	1.0	4025	2841
	sigma	0.997	0.006	0.986	1.008	1.0	4466	3074
2016	Intercept	0.001	0.008	-0.014	0.017	1.0	3808	2784
	dollars	0.091	0.008	0.075	0.107	1.0	4039	3079
	sigma	0.996	0.005	0.985	1.007	1.0	4824	2931
2017	Intercept	0.001	0.008	-0.014	0.017	1.0	3525	3064
	dollars	0.087	0.008	0.072	0.102	1.0	4740	2749
	sigma	0.996	0.006	0.985	1.008	1.0	5103	2953
2018	Intercept	0.001	0.008	-0.015	0.017	1.0	3962	2805
	dollars	0.097	0.008	0.082	0.114	1.0	4039	3203
	sigma	0.995	0.006	0.984	1.007	1.0	4631	3216
2019	Intercept	0.001	0.008	-0.014	0.016	1.0	4069	2763
	dollars	0.107	0.008	0.091	0.122	1.0	4011	2728
	sigma	0.994	0.006	0.983	1.006	1.0	4447	2835

Model predictions (2001-2019)

The posterior estimates of the piecewise model indicated that the relationship between happiness and income changed over time between 2002 and 2018. One implication of such a change is the disparity in happiness between income groups has increased. This will occur as more people fall below the change point over time and so are subject to the steep region of the function where income and happiness are strongly related. To directly examine the implications of our model for such inequities in happiness, we used the model to estimate or predict the happiness of two different income levels: one income level which was above the change point in 2001 but fell below it by 2019 ($50K/yr); and another level which always remained above the change points over the same period ($75K/yr). Figure S8 below presents the happiness levels (in SD units/year) for each income level as well as the difference in happiness between them (∆).

Figure S6. Happiness (SD units) at $50K/yr and $75K/yr from 2001-2019

Figure S6 legend: The difference (∆) in happiness between a household income of $50K per year and a household income of $75K per year has increased between 2002 and 2018.

Figure S6 shows the disparity in happiness between two fixed income levels, $50K/year (traversing the change points) and $75K/year (above the change points). In 2001, a household income of $50K/year achieved a near average level of happiness relative to everyone else that year, however by 2019 happiness had declined to lower than average levels relative to the population. By contrast, a household income of $75K/year enjoyed a higher than average level of happiness for almost the entire period. The difference (∆) in happiness between these two income levels doubled over the period. The increasing disparity in happiness between the rich and the poor implies that happiness has became more inequitable over the period.

In-sample fit statistics

Our primary objective was to understand the relationships between the parameters of our model, and so in-sample fit statistics such as R² are irrelevant for this objective. Moreover, our use of regularized priors will bias such in-sample estimates towards zero as regularization deliberately sacrifices in-sample variance for out-of-sample accuracy, which obviously hinders interpretation of the in-sample values. Nevertheless, some readers may find it useful to compare the in-sample fit statistics between the three different model types employed here: piecewise, linear and log-linear, as well as between the conditional and unconditional models. Because the usual fit statistics such as R² present a problem for Bayesian fits, as the variance of the predicted values can be larger than the variance of the data results in R² values greater than 1, we present the in-sample statistics from frequentist OLS model fits (without regularization or penalized likelihoods).

Table S4. In-sample fit statistics (R-squared)
year	Linear	Linear (conditional)	Log-linear	Log-linear (conditional)	Piecewise	Piecewise (conditional)
2001	0.014	0.117	0.013	0.117	0.017	0.118
2002	0.016	0.120	0.015	0.120	0.019	0.122
2003	0.019	0.118	0.020	0.119	0.022	0.119
2004	0.023	0.141	0.024	0.141	0.026	0.143
2005	0.021	0.134	0.022	0.135	0.025	0.137
2006	0.023	0.131	0.026	0.133	0.028	0.133
2007	0.020	0.132	0.026	0.135	0.028	0.137
2008	0.023	0.137	0.026	0.139	0.028	0.140
2009	0.022	0.142	0.023	0.144	0.026	0.145
2010	0.021	0.126	0.025	0.129	0.027	0.130
2011	0.023	0.123	0.023	0.124	0.028	0.127
2012	0.019	0.134	0.021	0.135	0.023	0.136
2013	0.019	0.140	0.020	0.141	0.025	0.143
2014	0.021	0.146	0.021	0.146	0.025	0.148
2015	0.020	0.146	0.019	0.146	0.023	0.148
2016	0.018	0.133	0.021	0.134	0.024	0.136
2017	0.018	0.145	0.019	0.146	0.023	0.148
2018	0.018	0.140	0.019	0.142	0.023	0.144
2019	0.018	0.133	0.016	0.132	0.021	0.135

A comparison of the relative differences in R² values between models indicates the piecewise model produced a slightly higher in-sample fit over the other two models, while the linear model tended to have the lowest R². The log-linear fit varied between both over the 19 years. As expected, adding covariates increased the R-squared in each case, but it did not change the relative difference between models.

Three things need to be observed when interpreting the R² values: 1) While the R² between income and happiness is low (~2% and ~14% in the unconditional and conditional models, respectively), the aggregate effect over the wider population (e.g., 25M in Australia) will be much larger; 2) Happiness (and to a lesser extent, income) is an imperfect and noisy measure so any effect of income on happiness is likely to be larger than that measured here; 3) Finally, happiness is measured on a subjective and arbitrary Likert scale so a small change in happiness at different points in the scale may represent a large change in some other functional/real-world outcome (e.g., risk of suicide). This is all to say that quantifying the real-world impact of small changes in variation reported here is difficult and speculative without further investigation.

Supplementary material: The increasing cost of happiness

R.W. Morris^1,2, N. Kettlewell^2,3,4 & N.Glozier^1,5